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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2023 • Question 4

Question

Math 6210 • Real Analysis Fall 2023 • Question 4

Let \(X=[0,1]\) and let \(\mu \) be a finite Borel measure on \(X\). Prove that \(\mu \) is regular, i.e. that for any Borel set \(A\subseteq X\) the following holds:

  1. \(\mu (A)=\sup \{\mu (K)\mid K\subseteq A,\ K\text { compact}\}\), and
  2. \(\mu (A)=\inf \{\mu (U)\mid U\supseteq A,\ U\text { open}\}\).

Note: The same statement holds for any compact metrizable space \(X\).

Raw LaTeX

Let \(X=[0,1]\) and let \(\mu\) be a finite Borel measure on \(X\). Prove that \(\mu\) is
regular, i.e. that for any Borel set \(A\subseteq X\) the following holds:
\begin{enumerate}

\item[(i)]
\(\mu(A)=\sup\{\mu(K)\mid K\subseteq A,\ K\text{ compact}\}\), and
\item[(ii)]
\(\mu(A)=\inf\{\mu(U)\mid U\supseteq A,\ U\text{ open}\}\).

\end{enumerate}

Note: The same statement holds for any compact metrizable space \(X\).

Topic Tags

Absolute Continuity and Singular Measures Borel Sets and Sigma-Algebras Compactness in Metric and Normed Spaces Measure Theory